Yes I had this exact problem.

On my car it had to do with a disconnected coolant temp switch. MB didn’t want to give you a gush of cold air when you start the car on a cold day so the blower will not turn on when the water is cold.

You can override it by pressing the buttons the way you describe. This was my experience.

I dumbheadedly replaced my button controller and the brain box behind the glove box and it didn’t fix it.

Turned out my coolant switch (located at the front of the engine near the thermostat and upper radiator hose) had become disconnected and the frayed end of the wire was intermittently grounding on the chassis of the car.

Hope that helps.

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79 300TD “Old Smokey” AKA “The Mistake” (SOLD)
82 240D stick shift 335k miles (SOLD)
82 300SD 300k miles
85 300D Turbodiesel 170k miles
97 C280 147k miles